The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$

Abstract

Let p be an odd prime and let k be a field of characteristic p. We provide a practical algebraic description of the representation ring of kSL2(Fp) modulo projectives. We then investigate a family of modular plethysms of the natural kSL2(Fp)-module E of the form ∇ ν Sym l E for a partition ν of size less than p and 0 ≤ l ≤ p − 2. Within this family we classify both the modular plethysms of E which are projective and the modular plethysms of E which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of E are replaced by kSL2(Fp)-modules of the form ∇ ν V , where V is a nonprojective indecomposable kSL2(Fp)-module and |ν| < p.